Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.81507.1057 Applied Algebra Survey of Graph Energies Survey of Graph Energies Gutman Ivan University Kragujevac, Serbia Furtula Boris State University of Novi Pazar, Novi Pazar; Serbia 01 12 2017 2 2 85 129 14 03 2016 14 03 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_46658.html

Let graph energy is a graph--spectrum--based quantity‎, ‎introduced in the 1970s‎. ‎After a latent period of 20--30 years‎, ‎it became a popular topic of research both‎ ‎in mathematical chemistry and in ``pure'' spectral graph theory‎, ‎resulting in‎ ‎over 600 published papers‎. ‎Eventually‎, ‎scores of different graph energies have‎ ‎been conceived‎. ‎In this article we provide the basic facts on graph energies‎, ‎in particular historical and bibliographic data.‎

Energy spectrum Graph
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.70534.1051 Applicable Analysis On Eccentricity Version of Laplacian Energy of a Graph On eccentricity version of Laplacian energy of a graph De Nilanjan Calcutta Institute of Engineering and Management 01 12 2017 2 2 131 139 17 12 2016 11 04 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_46665.html

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini and Panahbar [R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and other topological indices. J. Math. Nanosci. 2016, 6, 57-65], in this paper we investigate the eccentricity version of Laplacian energy of a graph G.

Eccentricity Eigenvalue energy (of graph) Laplacian energy topological index
1. T. Aleksic, Upper bounds for Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 435 - 439. 2. R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287 - 295. 3. K. C. Das, S. A. Mojallal, I. Gutman, On energy and Laplacian energy of bipartite graphs, Appl. Math. Comput. 273 (2016) 759 - 766. 4. N. De, New bounds for Zagreb eccentricity indices, Open J. Discrete Math. 3 (2013) 70 - 74. 5. J. B. Diaz, F. T. Metcalf, Stronger forms of a class of inequalities of G. PolyaG.Szego, and L. V. Kantorovich. Bull. Amer. Math. Soc. 69 (1963) 415-418. 6. G. H. Fath-Tabar, A. R. Ashrafi, Some remarks on Laplacian eigenvalues and Laplacian energy of graphs, Math. Commun. 15 (2010) 443 - 451. 7. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forsch. Graz 103 (1978) 22 pp. 8. I. Gutman, The energy of graph: old and new results, Algebraic combinatorics and applications, Springer, Berlin, (2001) 196 - 211. 9. I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29 - 37. 10. H. Liu, M. Lu, F. Tian, Some upper bounds for the energy of graphs, J. Math. Chem. 41 (2007) 45 - 57. 11. Z. Luo, J. Wu, Zagreb eccentricity indices of the generalized hierarchical product graphs and their applications, J. Appl. Math. 2014 Art. ID 241712, 8 pp. 12. R. Merris, Laplacian matrices of graphs: a survey, Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992) Linear Algebra Appl. 197/198 (1994) 143 - 176. 13. V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007) 1472 - 1475. 14. N. Ozeki, On the estimation of inequalities by the maximum, or minimum values, (Japanese) J. College Arts Sci. Chiba Univ. 5 (1968) 199 - 203. 15. G. Polya, G. Szego, Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions, Springer-Verlag, New York-Berlin (1972). 16. R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and other topological indices, J. Math. Nanosci. 6 (2016) 57 - 65. 17. R. Xing, B. Zhou, N. Trinajstic, On Zagreb eccentricity indices, Croat. Chem. Acta 84 (2011) 493 - 497. 18. B. Zhou, I. Gutman, T. Aleksic, A note on Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441 - 446.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.85687.1063 Applied Combinatorics On Relation between the Kirchhoff Index and Laplacian-Energy-Like Invariant of Graphs On Relation between the Kirchhoff Index and Laplacian-Energy-Like Invariant of Graphs Milovanovic Emina Faculty of Electronic Engineering, University of Nis, Nis, Serbia Milovanovic Igor Faculty of Electronic Engineering, University of Nis, Nis, Serbia Matejic Marjan Faculty of Electronic Engineering, University of Nis, Nis, Serbia 01 12 2017 2 2 141 154 11 05 2017 08 06 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_46678.html

Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ1 ≥ μ2 ≥...≥μn-1 >μn=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣi=1n-11/μi and LEL(G)=Σi=1n-1 √μi, respectively. In this paper we consider relationship between Kf(G) and LEL(G).

Kirchhoff index Laplacian-energy-like invariant Laplacian eigenvalues of graph
1. B. Arsić, I. Gutman, K. Ch. Das, K. Xu, Relations between Kirchhoff and Laplacian–energy–like invariant, Bull. Cl. Sci. Math. Nat. Sci. Math. 37 (2012) 59–70. 2. M. Biernacki, H. Pidek, C. Ryll-Nardzewski, Sur une inégalité entre des intégrales définies (French), Ann. Univ. Mariae Curie-Sklodowska. Sect. A. 4 (1950) 1–4. 3. B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Zagreb indices: Bounds and Extremal graphs, In: Bounds in Chemical Graph Theory – Basics, I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović (Eds.), Mathematical Chemistry Monographs, MCM 19, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac„ 2017, pp. 67–153. 4. B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17–100. 5. F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. 6. V. Cirtoaje, The best lower bound depended on two fixed variables for Jensen’s inequality with ordered variables, J. Inequal. Appl. (2010) Art. ID 128258,12 pp. 7. K. C. Das, K. Xu, I. Gutman, Comparison between Kirchhoff index and the Laplacian–energy–like invariant, Linear Algebra Appl. 436 (2012) 3661–3671. 8. K. Ch. Das, K. Xu, On relation between Kirchhoff index, Laplacian–energy– like invariant and Laplacian energy of graphs, Bull. Malays. Math. Sci. Soc. 39 (2016) S59–S75. 9. K. C. Das, On the Kirchhoff index of graphs, Z. Naturforsch 68a (2013) 531–538. 10. C. S. Edwards, The largest vertex degree sum for a triangle in a graph, Bull. London Math. Soc. 9 (1977) 203–208. 11. I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz 103 (1978) 22 pp. 12. I. Gutman, The energy of a graph: old and new results, In: Algebraic Combinators and Applications, Springer, Berlin, 2001, pp. 196–211. 13. I. Gutman, Editorial, Census of graph energies, MATCH Commun. Math. Comput. Chem. 74 (2015) 219–221. 14. I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92. 15. I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović (Eds.), Bounds in Chemical Graph Theory – Basics, Mathematical Chemistry Monographs, MCM 19, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, 2017. 16. I. Gutman, E. Milovanović, I. Milovanović, Bounds for Laplacian-type graph energies, Miskolc Math. Notes 16 (2015) 195–203. 17. I. Gutman, X. Li, (Eds.), Energies of Graphs – Theory and Applications, Mathematical Chemistry Monographs, MCM 17, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, 2016. 18. I. Gutman, B. Mohar, The quasi–Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996) 982–985. 19. I. Gutman, S. Radenković, S. Djordjević, I. Ž. Milovanović, E. I. Milovanović, Total Pi-electron and HOMO energy, Chem. Phys. Lett. 649 (2016) 148–150. 20. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total Pi-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. 21. I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. 22. D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81–95. 23. X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. 24. B. Liu, Y. Huang, Z. You, A survey on the Laplacian–energy–like invariant, MATCH Commun. Math. Comput. Chem. 66 (2011) 713–730. 25. J. Liu, J. Cao, X. F. Pan, A. Elaiw, The Kirchhoff index of hypercubes and related complex networks, Discrete Dyn. Nat. Soc. (2013) Art. ID 543189, 7 pp. 26. J. Liu, B. Liu, A Laplacian–energy–like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372. 27. A. Lupas, A remark on the Schweitzer and Kantorovich inequalities, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 383 (1972) 13–15. 28. I. Milovanović, I. Gutman, E. Milovanović, On Kirchhoff and degree Kirchhoff indices, Filomat 29 (2015) 1869–1877. 29. I. Ž. Milovanović, E. I. Milovanović, On some lower bounds of the Kirchhoff index, MATCH Commun. Math. Comput. Chem. 78 (2017) 169–180. 30. I. Ž. Milovanović, E. I. Milovanović, Bounds of Kirchhoff and degree Kirchhoff indices, In: Bounds in Chemical Graph Theory – Mainstreams, I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović, (Eds.), Mathematical Chemistry Monographs, MCM 20, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, 2017, pp. 93–119. 31. I. Ž. Milovanović, E. I. Milovanović, Remarks on the energy and the minimum dominating energy of a graph, MATCH Commun. Math. Comput. Chem. 75 (2016) 305–314. 32. I. MIlovanović, E. Milovanovioć, I. Gutman, Upper bounds for some graph energies, Appl. Math. Comput. 289 (2016) 435–443. 33. D. S. Mitrinović, P. M. Vasić, Analytic Inequalities, Springer Verlag, BerlinHeidelberg–New York, 1970. 34. S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124. 35. J. L. Palacios, Some additional bounds for the Kirchhoff index, MATCH Commun. Math. Comput. Chem. 75 (2016) 365–372. 36. S. Pirzada, H. A. Ganie, I. Gutman, On Laplacian–energy–like invariant and Kirchhoff index, MATCH Commun. Math. Comput. Chem. 73 (2015) 41–59. 37. S. Pirzada, H. A. Ganie, I. Gutman, Comparison between Laplacian–energy– like invariant and the Kirchhoff index, Electron. J. Linear Algebra 31 (2016) 27–41. 38. B. C. Rennie, On a class of inequalities, J. Austral. Math. Soc. 3 (1963) 442–448. 39. D. Stevanović, S. Wagner, Laplacian–energy–like invariant: Laplacian coefficients, extremal graphs and bounds, In: Energies of Graphs – Theory and Applications, I. Gutman, X. Li (Eds.), Mathematical Chemistry Monographs, MCM 17, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac, 2017, pp. 81–110. 40. H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17–20. 41. C. Woods, My Favorite Application Using Graph Eigenvalues: Graph Energy, Avaliable at http://www.math.ucsd.edu/ fan/teach/262/13/ 262notes/ Woods_Midterm.pdf 42. B. Zhou, N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. 455 (2008) 120–123. 43. B. Zhou, I. Gutman, T. Aleksić, A note on the Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441–446. 44. H. Y. Zhu, D. J. Klei, I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.57775.1038 Applied Algebra The Signless Laplacian Estrada Index of Unicyclic Graphs The Signless Laplacian Estrada Index of Unicyclic Graphs Ellahi Hamid Reza Department of Mathematics, Faculty of Science, University of Qom, Qom, I R Iran Nasiri Ramin Department of Mathematics, Faculty of Science, University of Qom, Qom, I R Iran Fath-Tabar Gholam Hossein Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kahsan, Kashan, Iran Gholami Ahmad Department of Mathematics, Faculty of Science, University of Qom, Qom, I R Iran 01 12 2017 2 2 155 167 14 07 2016 20 12 2016 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_46679.html

‎For a simple graph \$G\$‎, ‎the signless Laplacian Estrada index is defined as \$SLEE(G)=sum^{n}_{i=1}e^{q^{}_i}\$‎, ‎where \$q^{}_1‎, ‎q^{}_2‎, ‎dots‎, ‎q^{}_n\$ are the eigenvalues of the signless Laplacian matrix of \$G\$‎. ‎In this paper‎, ‎we first characterize the unicyclic graphs with the first two largest and smallest \$SLEE\$'s and then determine the unique unicyclic graph with maximum \$SLEE\$ among all ‎unicyclic graphs on \$n\$ vertices with a given diameter‎. ‎All extremal graphs‎, ‎which have been introduced in our results are also extremal with respect to the signless Laplacian ‎resolvent energy‎.

‎Signless Laplacian Estrada index‎ ‎unicyclic graphs‎ ‎extremal graphs‎ ‎diameter ‎signless Laplacian resolvent energy‎
1. N. Abreu, D. M. Cardoso, I. Gutman, E. A. Martins, M. Robbiano, Bounds for the signless Laplacian energy, Linear Algebra Appl. 435 (2011) 2365-2374. 2. S. K. Ayyaswamy, S. Balachandran,Y. B. Venkatakrishnan, I. Gutman, Signless Laplacian Estrada index, MATCH Commun. Math. Comput. Chem. 66(3) (2011) 785 - 794. 3. R. Binthiya, P. B. Sarasija, On the signless Laplacian energy and signless Laplacian Estrada index of extremal graphs, Appl. Math. Sci. 8 (2014) 193 -198. 4. A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastine, M. D. Safe, V. Trevisan, I. Gutman, Some Results for the (Signless) Laplacian Resolvent, MATCH Commun. Math. Comput. Chem. 77(1) (2017) 105 - 114. 5. D. Cvetković, P. Rowlinson, S. K. Simić, Eigenvalue bound for the signless Laplacian, Publ. Inst. Math. (Beograd) (N. S.) 81 (2007) 11 - 27. 6. D. Cvetković, P. Rowlinson, S. K. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155 - 171. 7. D. Cvetković, S. K. Simić, Towards a spectral theory of graphs based on the signless Laplacian I, Publ. Inst. Math. (Beograd) (N. S.) 85 (2009) 19 - 33. 8. E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002), Linear Algebra Appl. 373 (2003) 241 - 272. 9. H. R. Ellahi, G. H. Fath-Tabar, A. Gholami, R. Nasiri, On maximum signless Laplacian Estrada index of graphs with given parameters, Ars Math. Contemp. 11 (2016) 381 - 389. 10. Y. Z. Fan, Largest eigenvalue of a unicyclic mixed graph, Appl. Math. J. Chinese Univ. Ser. B 19 (2004) 140 - 148. 11. Y. Z. Fan, B. S. Tam, J. Zhou, Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order, Linear Multilinear Algebra 56 (2008) 381 - 397. 12. R. Nasiri, H. R. Ellahi, G. H. Fath-Tabar, A. Gholami, On maximum signless Laplacian Estrada index of graphs with given parameters II, arXiv:1410.0229. 13. R. Nasiri, H. R. Ellahi, G. H. Fath-Tabar, A. Gholami, T. Došlić, The signless Laplacian Estrada index of tricyclic graphs, Australas. J. Combin. In press. 14. R. Nasiri, H. R. Ellahi, A. Gholami, G. H. Fath-Tabar, A. R. Ashrafi, Resolvent Estrada and signless Laplacian Estrada indices of graphs, MATCH Commun. Math. Comput. Chem. 77(1) (2017) 157 - 176. 15. B. S. Tam, Y. Z. Fan, J. Zhou, Unoriented Laplacian maximizing graphs are degree maximal, Linear Algebra Appl. 429 (2008) 735 - 758.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.90820.1068 Applied Combinatorics More Equienergetic Signed Graphs More Equienergetic Signed Graphs ‎Ramane Harishchandra S. Department of Mathematics, Karnatak University, Dharwad - 580003, India Gundloor Mahadevappa M. Department of Mathematics, Karnatak University, Dharwad - 580003, India 01 12 2017 2 2 169 179 29 06 2017 17 07 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_49308.html

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction of unbalanced, noncospectral equieneregtic signed graphs on \$n geq 8\$ vertices.

Signed graph energy of a graph equienergetic graphs
1. B. D. Acharya, Spectral criterion for the cycle balance in networks, J. Graph Theory 4 (1980) 1–11. 2. R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287–295. 3. M. A. Bhat, S. Pirzada, On equienergetic signed graphs, Discrete Appl. Math. 189 (2015) 1–7. 4. V. Brankov, D. Stevanović, I. Gutman, Equienergetic chemical tress, J. Serb. Chem. Soc. 69 (2004) 549–553. 5. D. M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application, Academic Press, Inc., New York-London, 1980. 6. K. A. Germina, K. S. Hameed, T. Zaslavsky, On products and line graphs of signed graphs, their eigenvalues and energy, Linear Algebra Appl. 435 (2011) 2432–2450. 7. I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz 103 (1978) 1–22. 8. Y. Hou, L. Xu, Equienergetic bipartite graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 363–370. 9. G. Indulal, A. Vijayakumar, Equienergetic self-complementary graphs, Czechoslovak Math. J. 58 (2008) 911–919. 10. X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. 11. N. G. Nayak, Equienergetic net-regular signed graphs, Int. J. Contemp. Math. Sci. 9 (2014) 685–693. 12. H. S. Ramane, H. B. Walikar, Construction of equienergetic graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 203–210. 13. H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Equienergetic graphs, Kragujevac J. Math. 26 (2004) 5–13. 14. D. Stevanović, Energy and NEPS of graphs, Linear Multilinear Algebra 53 (2005) 67–74.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.101641.1081 Applied Combinatorics Seidel Signless Laplacian Energy of Graphs Seidel Signless Laplacian Energy of Graphs Ramane Harishchandra S. Karnatak University Gutman Ivan University Kragujevac, Serbia Patil Jayashri B. Hirasugar Institute of Technology Jummannaver Raju B. Karnatak University 01 12 2017 2 2 181 191 20 10 2017 06 11 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_53998.html

Let \$S(G)\$ be the Seidel matrix of a graph \$G\$ of order \$n\$ and let \$D_S(G)=diag(n-1-2d_1, n-1-2d_2,ldots, n-1-2d_n)\$ be the diagonal matrix with \$d_i\$ denoting the degree of a vertex \$v_i\$ in \$G\$. The Seidel Laplacian matrix of \$G\$ is defined as \$SL(G)=D_S(G)-S(G)\$ and the Seidel signless Laplacian matrix as \$SL^+(G)=D_S(G)+S(G)\$. The Seidel signless Laplacian energy \$E_{SL^+}(G)\$ is defined as the sum of the absolute deviations of the eigenvalues of \$SL^+(G)\$ from their mean. In this paper, we establish the main properties of the eigenvalues of \$SL^+(G)\$ and of \$E_{SL^+}(G)\$.

Seidel Laplacian eigenvalues Seidel Laplacian energy Seidel signless Laplacian matrix Seidel signless Laplacian eigenvalues Seidel signless Laplacian energy
1. N. Abreu, D. M. Cardoso, I. Gutman, E. A. Martins, M. Robbiano, Bounds for the signless Laplacian energy, Linear Algebra Appl. 435 (2011) 2365–2374. 2. M. Biernacki, H. Pidek, C. Ryll–Nardzewski, Sur une inégalité entre des intégrales définies, Univ. Maria Curie Skłodowska A4 (1950) 1–4. 3. B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17–100. 4. D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs – Theory and Application, Academic Press, New York, 1980. 5. D. Cvetković, P. Rowlinson, S. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171. 6. D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010. 7. D. Cvetković, S. K. Simić, Towards a spectral theory of graphs based on the signless Laplacian I, Publ. Inst. Math. (Beograd) 85 (2009) 19–33. 8. D. Cvetković, S. K. Simić, Towards a spectral theory of graphs based on the signless Laplacian II, Linear Algebra Appl. 432 (2010) 2257–2272. 9. D. Cvetković, S. K. Simić, Towards a spectral theory of graphs based on the signless Laplacian III, Appl. Anal. Discr. Math. 4 (2010) 156–166. 10. J. B. Diaz, F. T. Metcalf, Stronger forms of a class of inequalities of G.Pólya– G.Szegő and L.V.Kantorovich, Bull. Am. Math. Soc. 69 (1963) 415–418. 11. E. Ghorbani, On eigenvalues of Seidel matrices and Haemers conjecture, Designs, Codes Cryptography 84 (2017) 189–195. 12. R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discr. Math. 7 (1994) 221–229. 13. R. Grone, R. Merris, V. S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218–238. 14. I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz 103 (1978) 1–22. 15. I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92. 16. I. Gutman, B. Furtula, Survey of graph energies, Math. Interdisc. Res. 2 (2017) 85–129. 17. I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. 18. W. H. Haemers, Seidel switching and graph energy, MATCH Commun. Math. Comput. Chem. 68 (2012) 653–659. 19. X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. 20. R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197- 198 (1994) 143–176. 21. R. Merris, A survey of graph Laplacians, Linear Multilin. Algebra 39 (1995) 19–31. 22. B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk (Eds.), Graph Theory, Combinatorics and Applications, Wiley, New York, 1991, pp. 871–898. 23. P. Nageswari, P. B. Sarasija, Seidel energy and its bounds, Int. J. Math. Anal. 8 (2014) 2869–2871. 24. M. R. Oboudi, Energy and Seidel energy of graphs, MATCH Commun. Math. Comput. Chem. 75 (2016) 291–303. 25. N. Ozeki, On the estimation of inequalities by maximum and minimum values, J. Coll. Arts Sci. Chiba Univ. 5 (1968) 199–203 (in Japanese). 26. G. Pólya, G. Szegő, Problems and Theorems in Analysis, Series, Integral Calculus, Theory of Functions, Springer, Berlin, 1972. 27. M. R. Rajesh Kanna, R. P. Kumar, M. R. Farahani, Milovanović bounds for Seidel energy of a graph, Adv. Theor. Appl. Math. 10 (2016) 37–44. 28. H. S. Ramane, M. M. Gundloor, S. M. Hosamani, Seidel equienergetic graphs, Bull. Math. Sci. Appl. 16 (2016) 62–69. 29. H. S. Ramane, I. Gutman, M. M. Gundloor, Seidel energy of iterated line graphs of regular graphs, Kragujevac J. Math. 39 (2015) 7–12. 30. H. S. Ramane, R. B. Jummannaver, I. Gutman, Seidel Laplacian energy of graphs, Int. J. Appl. Graph Theory 1 (2017) 74–82.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.101675.1082 Applied Algebra Eigenvalues of the Cayley Graph of Some Groups with respect to a Normal Subset Eigenvalues of the Cayley Graph of Some Groups with respect to a Normal Subset Jalali-Rad Maryam Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran 01 12 2017 2 2 193 207 21 08 2017 06 11 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_53999.html

‎‎Set X = { M11‎, ‎M12‎, ‎M22‎, ‎M23‎, ‎M24‎, ‎Zn‎, ‎T4n‎, ‎SD8n‎, ‎Sz(q)‎, ‎G2(q)‎, ‎V8n}‎, where M11‎, M12‎, M22‎, ‎M23‎, ‎M24 are Mathieu groups and Zn‎, T4n‎, SD8n‎, ‎Sz(q)‎, G2(q) and V8n denote the cyclic‎, ‎dicyclic‎, ‎semi-dihedral‎, ‎Suzuki‎, ‎Ree and a group of order 8n presented by                                      V8n = < a‎, ‎b | a^{2n} = b^{4} = e‎, ‎ aba = b^{-1}‎, ‎ab^{-1}a = b>,respectively‎. ‎In this paper‎, ‎we compute all eigenvalues of Cay(G,T)‎, ‎where G in X and T is minimal‎, ‎second minimal‎, ‎maximal or second maximal normal subset of G{e} with respect to its size‎. ‎In the case that S is a minimal normal subset of G{e}‎, ‎the summation of the absolute value of eigenvalues‎, ‎energy of the Cayley graph‎, ‎are evaluated‎.

Simple group‎ ‎Cayley graph‎ ‎eigenvalue‎ ‎energy
1. R. C. Alperin, Rational subsets of finite groups, Int. J. Group Theory 3(2) (2014) 53–55. 2. R. C. Alperin, B. L. Peterson, Integral sets and Cayley graphs of finite groups, Electron. J. Combin. 19 (2012) #P44 1–12. 3. H. Behravesh, Quasi-permutation representation of Suzuki group, J. Sci. I. R. Iran 10(1) (1999) 53–56. 4. N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974. 5. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Clarendon, Oxford, 1985. 6. M. R. Darafsheh, N. S. Poursalavati, On the existence of the orthogonal basis of the symmetry classes of tensors associated with certain groups, SUT J. Math. 37(1) (2001) 1–17. 7. P. Diaconis, M. Shahshahani, Generating a Random Permutation with Random Transpositions, Z. Wahr. Verw. Gebiete 57(2) (1981) 159–179. 8. M. Ghorbani, On the eigenvalues of normal edge-transitive Cayley graphs, Bull. Iranian Math. Soc. 41(1) (2015) 101–107. 9. I. Gutman, A graph theoretical study of conjugated systems containing a linear polyence fragment, Croat. Chem. Acta 48(2) (1976) 97–108. 10. I. Gutman, The energy of a graph, 10. Steierm¨arkisches Mathematisches Symposium (Stift Rein, Graz, 1978). Ber. Math.-Statist. Sekt. Forsch. Graz 103 (1978) 22 pp. 11. M. Hormozi, K. Rodtes, Symmetry classes of tensors associated with the semi-dihedral groups SD8n, Colloq. Math. 131(1) (2013) 59–67. 12. I. M. Isaacs, Character Theory of Finite Groups, Dover, New-York, 1976. 13. G. James, M. Liebeck, Representations and Characters of Groups, Cambridge Univ. Press, London-New York, 1993. 14. M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(1) (2003) 1–20. 15. R. Ree, A family of simple groups associated with the simple Lie algebra of type (G2), Amer. J. Math. 83 (1961) 432–462. 16. M. Suzuki, On a class of doubly transitive groups, Annals Math. 75(1) (1962) 105–145. 17. The GAP Team, GAP-Groups, Algorithms and Programming, Lehrstul D F¨ur Mathematik, RWTH, Aachen, 1995. 18. H. N. Ward, On Ree’s series of simple groups, Trans. Amer. Math. Soc. 121(1) (1966) 62–89. 19. P. -H. Zieschang, Cayley graphs of finite groups, J. Algebra 118 (1988) 447–454.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2017.106176.1084 Applied Algebra Laplacian Sum-Eccentricity Energy of a Graph Laplacian Sum-Eccentricity Energy of a Graph Sharada Biligirirangaiah Mysore University, Mysore, India Sowaity Mohammad Issa Mysore University, Mysore, India Gutman Ivan University Kragujevac, Serbia 01 12 2017 2 2 209 219 19 11 2017 12 12 2017 Copyright © 2017, University of Kashan. 2017 http://mir.kashanu.ac.ir/article_54000.html

We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions for the existence of such graphs are established.

Sum-eccentricity eigenvalues sum-eccentricity energy Laplacian sum-eccentricity matrix Laplacian sum-eccentricity energy
1. P. G. Bhat, S. D’Souza, Color signless Laplacian energy of graphs, AKCE Int. J. Graphs Comb. 14 (2017) 142–148. 2. Ş. B. Bozkurt, A. D. Güngör, I. Gutman, Randić spectral radius and Randić energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 321–334. 3. Ş. B. Bozkurt, A. D. Güngör, I. Gutman, A. S Çevik, Randić matrix and Randić energy, MATCH Commun. Math. Comput. Chem. 64 (2010) 239–250. 4. F. Buckley, F. Harary, Distance in Graphs, Addison–Wesley, Redwood, 1990. 5. A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastine, M. D. Safe, V. Trevisan, I. Gutman, Some results for the (signless) Laplacian resolvent, MATCH Commun. Math. Comput. Chem. 77 (2017) 105–114. 6. D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010. 7. K. C. Das, S. Sorgun, K. Xu, On Randić energy of graphs, in: I. Gutman, X. Li (Eds.), Graph Energies – Theory and Applications, Univ. Kragujevac, Kragujevac, 2016, pp. 111–122. 8. N. De, On eccentricity version of Laplacian energy of a graph, Math. Interdisc. Res. 2 (2017) 131–139. 9. E. Estrada, The ABC matrix, J. Math. Chem. 55 (2017) 1021–1033. 10. I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz 103 (1978) 1–22. 11. I. Gutman, B. Furtula, Survey of graph energies, Math. Interdisc. Res. 2 (2017) 85–129. 12. I. Gutman, X. Li (Eds.), Graph Energies – Theory and Applications, Univ. Kragujevac, KragujevaC, 2016. 13. I. Gutman, B. Zhou, Laplacian energy of a graph, Lin. Algebra Appl. 414 (2006) 29–37. 14. S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, QSPR analysis of certain graph theoretical matrices and their corresponding energy, Appl. Math. Nonlin. Sci. 2 (2017) 131–150. 15. N. Jafari Rad, A. Jahanbani, I. Gutman, Zagreb energy and Zagreb Estrada index of graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 371–386. 16. X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. 17. B. J. McClelland, Properties of the latent roots of a matrix: The estimation of Pi-electron energies, J. Chem. Phys. 54 (1971) 640–643. 18. Z. Meng, B. Xiao, High–resolution satellite image classification and segmentation using Laplacian graph energy, Proceeding of Geoscience and Remote Sensing Symposium (IGARSS), Vancouver, 2011, pp. 605–608. 19. J. K. Merikoski and R. Kumar, Inequalities for spreads of matrix sums and products, Appl. Maths. E-notes 4 (2004) 150–159. 20. H. S. Ramane, I. Gutman, J. B. Patil, R. B. Jummannaver, Seidel signless Laplacian energy of graphs, Math. Interdisc. Res. 2 (2017) 181–191. 21. H. S. Ramane, R. B. Jummannaver, I. Gutman, Seidel Laplacian energy of graphs, Int. J. Appl. Graph Theory 1(2) (2017) 74–82. 22. D. S. Revankar, M. M. Patil, H. S.Ramane, On eccentricity sum eigenvalue and eccentricity sum energy of a graph, Ann. Pure Appl. Math. 13 (2017) 125–130. 23. J. M. Rodríguez, J. M. Sigarreta, Spectral properties of geometric–arithmetic index, Appl. Math. Comput. 277 (2016) 142–153. 24. B. Sharada, M. Sowaity, On the sum-eccentricity energy of a graph, TWMS J. Appl. Eng. Math., in press. 25. Y. Z. Song, P. Arbelaez, P. Hall, C. Li, A. Balikai, Finding semantic structures in image hierarchies using Laplacian graph energy, in: K. Daniilidis, P. Maragos, N. Paragios (Eds.), Computer Vision – CECV 2010 (European Conference on Computer Vision, 2010), Part IV , Springer, Berlin, 2010, pp. 694–707. 26. M. Sowaity and B. Sharada, The sum-eccentricity energy of a graph, Int. J. Rec. Innovat. Trends Comput. Commun. 5 (2017) 293–304. 27. H. Zhang, X. Bai, H. Zheng, H. Zhao, J, Zhou, J. Cheng, H. Lu, Hierarchical remote sensing image analysis via graph Laplacian energy, IEEE Geosci. Remote Sensing Lett. 10 (2013) 396–400.